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Geometry Difficulty 8.2 Shortlist Prove it Baltic Way

Let's call the intersection of two segments almost perfect if for each of them the length of the segment is at least $2021$ times the distance between its midpoint and the intersection point.
Prove that there exists a closed broken line that intersects each of its segments at least once and for which all its intersections are almost perfect.

Solution

Consider two equilateral triangles with common centre and parallel sides. The closed broken line $A_1B_2C_1A_2B_1C_2$ has three intersections, because of symmetry we will consider only one.
Assume that $A_1B_2$ intersects $A_2B_1$ in point $X$ . Triangles $A_1B_1X$ and $A_2B_2X$ are similar therefore $\frac{A_1X}{XB_2} = \frac{A_1B_1}{A_2B_2}$ . It is enough to choose initial triangles of close enough size $1 < \frac{A_1B_1}{A_2B_2} < \frac{2022}{2021}$ to make the intersection $X$ almost perfect. $\square$

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Source: MathNet, licensed CC-BY-4.0. Statement reproduced verbatim; metadata (topic, difficulty) added by this project.